Problem

Source: Iran 3rd round 2011-Number Theory exam-P3

Tags: algebra, polynomial, Ring Theory, modular arithmetic, number theory proposed, number theory



$P(x)$ and $Q(x)$ are two polynomials with integer coefficients such that $P(x)|Q(x)^2+1$. a) Prove that there exists polynomials $A(x)$ and $B(x)$ with rational coefficients and a rational number $c$ such that $P(x)=c(A(x)^2+B(x)^2)$. b) If $P(x)$ is a monic polynomial with integer coefficients, Prove that there exists two polynomials $A(x)$ and $B(x)$ with integer coefficients such that $P(x)$ can be written in the form of $A(x)^2+B(x)^2$. Proposed by Mohammad Gharakhani